In many areas of the country, the sidewalk vs. road debate is moot. Laws tend to lean toward keeping cyclists off of walkways completely, especially in congested city centers. Simply put, sidewalks are for feet, not wheels.
State laws on the issue do vary, though, and states almost always leave the ultimate decision to localities. Local governments, in turn, have typically ruled against letting cyclists over the age of 13 use sidewalks, making it illegal and a ticketable offense. In these cases, cyclists can't use crosswalks, either -- any pedestrian travel space is off-limits to bikes.
Legal definitions reveal a lot in this regard. A bicycle is defined as a vehicle -- a human-powered (or hybrid-powered), typically two-wheeled vehicle, but a vehicle nonetheless. As such, a cyclist is a driver. Bicycles and their drivers, like cars and their drivers, belong in areas designated for the higher speeds at which those vehicles travel, and bike drivers have to follow all of the laws that car drivers do. In these terms, a bicycle driving on the sidewalk would be no different from a car driving on the sidewalk.
In areas where cyclists can use pedestrian pathways, on the other hand, their legal status typically varies with their location. When a bicycle is travelling on the road, the bike is a vehicle, the cyclist is a driver, and that driver has to follow the same traffic laws that cars do -- stopping at stop signs, signaling and obeying traffic lights, for instance. When, on the other hand, a cyclist is riding on the sidewalk, he or she is a pedestrian and must follow all the rules and regulations that pedestrians have to follow. At a signaled crosswalk, that cyclist has to wait for the hand to light up before proceeding just like the guy travelling on foot.
And even where it's legal, there are conditions intended to make sidewalk-sharing as safe as possible for pedestrians, including limiting bike speed to walking speed, setting rules for alerting walkers to the intention to pass, and mandatory bike-walking when a sidewalk is busy enough to make colliding with a pedestrian reasonably likely.
Sidewalk-cycling rules and regulations, then, are typically focused on protecting those on foot, not those on wheels. After all, the rider-bike combination is heavier than a pedestrian and is capable of moving at much faster speeds. From this, one might infer that cyclists face no danger on the sidewalk -- or at least far less danger than on the street.
This inference, it turns out, is quite problematic ...